LABELLING THE CHARACTER TABLES OF SYMMETRIC AND ALTERNATING GROUPS

Let X be a character table of the symmetric group S_n. It isshown that unless n = 4 or n = 6, there is a unique way to assignpartitions of n to the rows and columns of X so that for all and , X is equal to (), the value of the irreducible characterof S_n labelled by on elements of cycle type . Analogous resultsare proved for alternating groups, and for the Brauer charactertables of symmetric and alternating groups.     

SIMULTANEOUS DIOPHANTINE APPROXIMATION BY SQUARE-FREE NUMBERS

Let ₁, ...,_r R be ‘not very well approximable’,for example, Q-linearly independent real algebraic numbers.Then there are infinitely many positive square-free integersn such that ||n_i|| << n^–(2/3r)+(1 i r), where||·|| denotes distance to the nearest integer.     

MOTIVIC INVARIANTS OF ARTIN STACKS AND 'STACK FUNCTIONS'

An invariant of quasiprojective -varieties X with values ina commutative ring is motivic if (X) = (Y) + (X\ Y) for Y closedin X, and (X x Y) = (X)(Y). Examples include Euler characteristics and virtual Poincaré and Hodge polynomials. We firstdefine a unique extension ' of to finite type Artin -stacks, which is motivic and satisfies '([X/G]) = (X)/(G) when X is a -variety, G a special -groupacting on X, and [X/G] is the quotient stack. This only worksif (G) is invertible in for all special -groups G, which excludes = as (_m) = 0. But we can extend the construction to get roundthis. Then we develop the theory of stack functions on Artin stacks.These are a universal generalization of constructible functionson Artin stacks. There are several versions of the construction:the basic one , and variants ‘twisted’ by motivic invariants. We associate a -vector space or a -module to each Artin stack , with functorial operations of multiplication, pullbacks * and pushforwards _*under 1-morphisms ;, and so on. They will be important tools in the author's series on ‘Configurationsin abelian categories’.     

CONFORMAL MEASURES ASSOCIATED TO ENDS OF HYPERBOLIC n-MANIFOLDS

Let be a non-elementary Kleinian group acting on the closedn-dimensional unit ball and assume that its Poincaréseries converges at the exponent . Let M be the -quotient ofthe open unit ball. We consider certain families = {E₁, ...,E_p} of open subsets of M such that is compact. The sets E_i are the ends of M and is a completecollection of ends for M. We associate to each end E an -conformalmeasure such that the measures corresponding to different endsare mutually singular if non-trivial. Additionally, each -conformalmeasure for on the limit set () of can be written as a sumof such conformal measures associated to ends E . In dimension3, our results overlap with some results of Bishop and Jones(The law of the iterated logarithm for Kleinian groups, Cont.Math. 211 (1997), 17–50.).     

ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN

In this paper, we consider the following semilinear ellipticproblem: where f(x, t) tends to p(x) and q(x) L(^N), respectively, ast 0 and t +. We prove that there exist two numbers l and Lwith L < l such that problem (P) has at least one positivesolution for (– l, –L) and has no positive solutionfor all [–l,–L]. The existence and non-existenceof positive solutions for problem (P) at = –l and =–L are also discussed.     

THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Let (n) denote the Fourier coefficients of cusp forms or thenumber of divisors of n. Estimates of the type are shown, uniformly in q X. The methods canbe extended to other arithmetic functions, for example, thenumber of representations of n as a sum of two squares or k-freenumbers. As an application, sums of the type _{n X}(n) (n) forany q-periodic function can be estimated non-trivially.     

Elliptic Problems Involving Phase Boundaries Satisfying a Curvature Condition

Two theorems related to equilibrium free-boundary problems arepresented. One arises as a time-independent solution to thephase-field equations. The other is the relevant time-independentproblem for the Stefan model, modified for the surface tensioneffect. It also serves as a preliminary result for the phase-fieldformulation. Under appropriate conditions, we prove that, givenan appropriate positive constant and a smooth function u: R;,where is an annular domain in R², there exists a curve suchthat u(x)=—K(x) for all x , where K is the curvature.Using this result, we prove the existence of solutions to O=²+ ?(—³) + 2u that have a transition layer behaviour (from=—1 to =+1) for small and make the transition on thecurve . This proves there exist solutions to the phase fieldmodel that satisfy a Gibbs-Thompson relation.     

Arithmetical Properties of Powers of Algebraic Numbers

We consider the sequences of fractional parts {ⁿ}, n = 1, 2,3,..., and of integer parts [ⁿ], n = 1, 2, 3,..., where isan arbitrary positive number and > 1 is an algebraic number.We obtain an inequality for the difference between the largestand the smallest limit points of the first sequence. Such aninequality was earlier known for rational only. It is alsoshown that for roots of some irreducible trinomials the sequenceof integer parts contains infinitely many numbers divisibleby either 2 or 3. This is proved, for instance, for , n = 1, 2, 3,.... The fact that thereare infinitely many composite numbers in the sequence of integerparts of powers was proved earlier for Pisot numbers, Salemnumbers and the three rational numbers 3/2, 4/3, 5/4, but nosuch algebraic number having several conjugates outside theunit circle was known. 2000 Mathematics Subject Classification11J71, 11R04, 11R06, 11A41.     

COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN-BILU THEOREM

We note a link between combinatorial results of Bollobásand Leader concerning sumsets in the grid, the Brunn–Minkowskitheorem and a result of Freiman and Bilu concerning the structureof sets A with small doubling. Our main result is the following. If > 0 and if A is a finitenon-empty subset of a torsion-free abelian group with |A + A| K|A|, then A may be covered by e^KO(1) progressions of dimension log ₂ K + and size at most |A|.     

Quasi-Affinity in certain Classes of Operators

The family of operators S + V (, C, Re > 0), where V isan injective S-Volterra operator (that is, [S, V[ = V²) and— A — V^–1 generates a uniformly bounded C₀-semigroup,is studied in the context of similarity and of the weaker quasi-affinityrelation. It is shown that S is similar to S + V for all , C,Re > 1, and is a quasi-affine transform of S + tV for allt 0 and 0 < < 1.     

A Variation on a Theorem of Galvin and Hajnal

Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =₍₎₊₍₎, < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product_</I admits a cofinal -scale of ordinal functions. Then ₊, where =||||_I is the I-norm of .     

Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball

Weighted Triebel–Lizorkin and Besov spaces on the unitball B^d in ^d with weights w_µ(x)=(1–|x|²)^µ–1/2,µ0, are introduced and explored. A decomposition schemeis developed in terms of almost exponentially localized polynomialelements (needlets) {}, {} and it is shown that the membershipof a distribution to the weighted Triebel–Lizorkin orBesov spaces can be determined by the size of the needlet coefficients{f, } in appropriate sequence spaces.     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

Lp Lq Estimates for Parabolic Systems in Non-Divergence Form with VMO Coefficients

Consider a parabolic NxN-system of order m on ⁿ with top-ordercoefficients a VMOL. Let 1 < p, q < and let be a Muckenhouptweight. It is proved that systems of this kind possess a uniquesolution u satisfying whereAu = _||m a Du and J = [0,). In particular, choosing = 1, therealization of A in L^p(ⁿ)^N has maximal L^p – L^q regularity.     

Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

Herman Rings and Arnold Disks

For (,a) C* x C, let f_,a be the rational map defined by f_,a(z)= z² (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f_,a has a fixed Hermanring with rotation number (we consider that (e²ⁱ,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e²ⁱ ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e²ⁱ z(1+z). As a consequence, we show that for a (0,1/3), if f_l,a has afixed Herman ring with rotation number and if m_a is the modulusof the Herman ring, then, as a0, we have e m_a=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e^(a/2)(z-1/z).     

ON MINIMAL SURFACES WHOSE GAUSS MAP IS INVARIANT BY A HOLOMORPHIC FOLIATION

We study real minimal surfaces : M^{2 4} under the hypothesisthat the holomorphic Gauss map of the immersion is invariantby a holomorphic foliation with singularities. We give a sortof Huber–Osserman theorem regarding the algebraicity ofthe holomorphic Gauss map.     

ON THE LEAST n WITH {chi}(n) != 1

Suppose that a > 1. By using a method of Linnik employinghis Large Sieve one may derive the following result. The numberof primitive Dirichlet characters with conductor Q such that(n) = 1 for all n (log Q)^a is O(Q^2/a+). We improve the exponent2/a for a > 2 by using a refined version by Heath-Brown ofthe Halasz–Montgomery ‘Large Values method’.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.